Verschuere Applications to Synthetic CDO Pricing86

7/31/2019 Verschuere Applications to Synthetic CDO Pricing86

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On Copulas and their Application to CDOPricing

Benjamin Verschuere

January 2006 (First draft: November 2005)

Abstract

In this paper we present a factor approach combined with copula functions to pricetranches of synthetic Collateralized Debt Obligation (CDO) having totally inhomo-geneous collateral (the obligors in the CDO pool have different spreads and differentnotional) . While a CDO is a portfolio of defaultable fixed income products, copulasare functions which link univariate distributions together to build a multivariate distri-bution function. The attractiveness of copulas lies in their flexibility to simulate or fitdependant variables and their ability to provide scale invariant measures of associationbetween random variables. When pricing a synthetic CDO, the copula function will beused in conjunction with the factor approach to model the obligors risk neutral jointdefault probabilities. This paper will, in one hand, interest people already familiarwith the copula theory but unfamiliar to their application for the pricing of (correla-tion derivatives) financial instruments; while on the other hand, it will be relevant forpeople familiar with the credit derivatives products willing to know more about thecopula functions.

Keywords: Copulas Function, Dependance Concept, Factor Modeling, Credit Derivatives,CDO.

Ph.D student in statistics at the University of Toronto. 100 St. George St., Toronto, Ontario M5S

3G3, Canada (E-mail: [email protected]). This paper is the master thesis realized for thecompletion of my master in statistics at the University of Chicago. First and foremost I would like tothank my advisor Per Mykland; Alex Kreinin and Philip Schonbucher for the informations they providedme; Xiaofeng Shao and Jie Yang for their help; Oli Atlason for the many interesting discussions and theparticipants at University of Chicago master in statistics seminar. Comments and suggestions are welcome.The usual disclaimer applies.

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Copula are functions that link multivariate distributions to their one-dimensionalmargins

Schweizer (1991)

(...) dependance functions are the real key to the study of multivariate distrib-utions

Deheuvels (1980)

1 IntroductionThe aim of this paper is to bring a unified and general framework for the pricing of CDO

with factor copulas by introducing the theory behind the copula functions and applying it tothe pricing of credit derivatives. This paper does not claim to bring a new approach to theproblem but nevertheless presents an introduction to the concept of copula and how theycan, in a simple setting, be applied to the pricing of CDO.The first section of this paper will be devoted on the copula theory, while the second one willfocus on the notion of dependance measures and the third section will provide an overview ofsome copulas families. The fourth section concerns the default modeling and will end by anapplication of the copulas function to the pricing of totally inhomogeneous synthetic CDO.

2 Mathematical and Statistical definition of Copulas

These citations above provide us with the scope of this section, defining what are copulasfunctions, what are their properties and how they can help us in the study of multivariatedistributions. For a short introduction about the copula the reader should to refer to Genestand McKay (1986b), for an extensive review on that topic the reader should refer to Nelsen(1998) Joe (1997) and Schweizer (1991) or for their application in risk management to Roggeand Schonbucher (2003) or Embrechts, Lindskog and Mc Neil (2003). The first part of this

section will be devoted to some notions in statistics, the interpretation of copula in themathematical sense and then in term of random variable while finally concluding with someimportant copula properties.

Let us first refresh some important concepts. We will start by the notion of distribution andjoint distribution function since they are at the cornerstone of the copula theory.

Definition 2.1 A distribution function is a function F with domain R such that

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1. F is nondecreasing

2. F(

) = 0, F(+

) = 1

Definition 2.2 A joint distribution function is a function H with domain R2 such that

1. H is 2-increasing

2. H(x,)= H(, y)= 0, H(+, +) = 1

In the last definition, the first condition simply stipulate that for every x1 x2, y1 y2 andH(x1, y1) H(x2, y2). While the second condition state that H should be grounded. Let usnow move to the main topic of this section by defining the notion of copula in the bivariate

case (for the tractability of notation)

Definition 2.3 A two dimensional copula (2-copula), C, is a real function defined on I2 =[0, 1] [0, 1], with range I = [0, 1], such that

1. For every(u, v) of I2, C(u, 0) = C(0, v) = 0 and C(u, 1) = u, C(1, v) = v

2. For every rectangle [u1, u2] [v1, v2] in I2 with u1 u2 and v1 v2,C(u2, v2) C(u2, v1) C(u1, v2) + C(u1, v1) 0

While once again, the first part of the definition states the 2-copula is a grounded functionthe latter ensure the volume engendered by the rectangle [u1, u2] [v1, v2] (or the 2-copula)is never negative. The copula can then be interpreted as a joint distribution. For the morepragmatic inclined, since the 2-copula can be seen as a volume in I, the form of the copulais the shape of a skewed continuous surface on the unit square which has vertices belongs tothe unit cube. The most important theorem in the copula theory is the following,

Theorem 2.1 Sklars Theorem Let H be a bivariate joint distribution function with mar-gins F and G. Then there exist a copula, C, such that for all x,y in R

H(x, y) = C(F(x), G(y))

Furthermore, if F and G are continuous then C is unique; otherwise C is uniquely determinedon RanFRanG. Conversely, if C is a copula and F and G are distribution function, thenH as defined by the previous expression is a joint distribution function with margins F andG.

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The ingenuity of this theorem lies in the fact that a joint distribution can be decomposedinto marginal distributions provided a function, the copula, which link them together.One example is to set F and G to be exponential margins and H to be the Gaussian linkfunction with a given covariance matrix, R and lets call this bivariate distribution Z. Asexplained in Schonbucher (2003) in order to construct this copula one has to sample a vectorof observation X from a multivariate Gaussian distribution with covariance matrix R. Thentransform this vector X into a vector U by setting ui = (xi), finally compute the vector Ywhere yi = ln(ui). Then yi follows the distribution Z.

If we now define F1 as the generalized inverse of the distribution function F such thatF1(t) = inf{x R | F(x) t} for all t in [0,1] using by the convention inf = . Wecan now extend the previous theorem as follows.

Corollary 2.2 LetH be a joint distribution function with continuous margins F andG andC a copula as previously defined then for every u and v in I

C(u, v) = H(F1(u), G1(v))

This last representation of the copula function will be particularly useful in the next section.

We now move the definition of copula in term of random variable, the following theoremcomes from the Sklar Theorem

Theorem 2.3 Let X and Y be random variable with, respectively, marginal distributionfunctions F and G and joint distribution function H, then there exist a copula C such thatdefinition 1.1 hold. If this F and G are continuous, C is unique. Otherwise C is determinedon RanF RanG.

We will call the copula of X and Y CX,Y. In the case of independence of X and Y we havethe following theorem.

Theorem 2.4 If X and Y are are said to be continuous independent random variables, iftheir joint distribution H is defined as

H(x, y) = F(x)H(y)then the copula for these independent variables is called the product copula, C

C = F(x)G(y)

After introducing the product copula one might naturally think about the existence of somelower and upper bounds around the product copula according to the dependence structurebetween the two random variables X and Y. These bounds are referred as the Frechet-Hoeffding bonds for joint distribution functions of random variables.

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Theorem 2.5 Frechet-Hoeffding Let X and Y be random variables with, respectively,marginal distribution functions F and G and with joint distribution H, then for all, x andy in R

max(F(x) + G(y) 1, 0) H(x, y) min(F(x), G(y))and

1. Y is a.s an increasing function of X iif H(x, y) = min(F(x), G(y))

2. Y is a.s an decreasing function of X iif H(x, y) = max(F(x) + G(y) 1, 0)

While the independence condition was previously stated.More generally, by the Theorem 1.2 and Theorem 1.5 we have the following corollary

Corollary 2.6 If u and v are uniform random variable, for all u and v in I

W = max(u + v 1, 0) C(u, v) min(u, v) = MProof: Lets take (u, v) an arbitrary point from DomC. Since C(u, v) C(1, v) = v and

C(u, v) C(u, 1) = u sot that C(u, v) min(u, v).To prove the left hand side, we use the fact that C(u,v)= VC([u, 1] [v, 1]) + u + v 1 andsince VC([u, 1] [v, 1]) 0 we have that C(u, v) max(u + v 1, 0).

It is worthwhile to be noted that the Frechet-Hoefding bounds are copulas in the bivariatecase but W is generally not a copula in the multivariate case (while M does). If we now look

at the shape of the surface defined by the copula, we can say that this shape is bounded bythe two Frechet-Hoeffding bounds which are functions in the unit cube.

To conclude the first part of this section, we introduce the second most interesting propertyof the copula (after the Sklar Theorem) : their invariance to strictly increasing transformationand predictable behavior for more general strictly monotone transformations 1.

Theorem 2.7 Let X,Y be continuous random variable with copula CX,Y. Let and bestrictly increasing functions on RanGRanF, respectively, then C(X),(Y) = CX,Y. So thatCx,y is invariant under strictly increasing transformation of X and Y.

Proof: Let X and Y have distributions function F and G and let (X) and (Y) havedistribution functionL and M. if we now set as (monotonic) increasing function, we havethe the following expressions for the transformation of marginal distribution of X,

L(x) = P{(X) x} = P{X 1(x)} = F(1(x))1From Wolff and Scheizer(1981) (...) for us the true importance of copulas lies in a combination of

Sklars Theorem (...) and the fact that under a.s strictly increasing transformation of X and Y the copulais invariant while the margins may be changed at will.(...) Hence the study of of rank statistics-insofaras it is the study of properties invariant under such transformation-may be characterized as the study ofcopulas-invariant properties.

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When looking at the transformation of the copula, we have

C(X),(Y)(L(X), M(Y)) = P{(X)

x,(Y)

y}

= P{X 1(x), Y 1(y)}= CXY(

1(x),1(y))

= CXY(L(X), M(Y))

When monotone transformations (but not increasing) are applied to copulas, the previousresult does not hold anymore but nonetheless we can still make the following statementabout the copula behavior.

Theorem 2.8 Let X,Y be continuous random variable with copula CX,Y. Let and bestrictly monotone function on RanGRanF, respectively

1. If is strictly increasing and is strictly decreasing then

C(X),(Y) = u CX,Y(u, 1 v)

2. If is strictly increasing and is strictly decreasing then

C(X),(Y) = v CX,Y(1 u, v)

3. If and are both strictly decreasing then

C(X),(Y) = u + v 1 + CX,Y(1 u, 1 v)

Proof: LetX, Y follows the distributions F and G while (X) and(X) follows the distri-bution L and M, with and monotone function so that

C(X),(X)(L(X), M(Y)) = P{(X) x,(Y) y}

So when is a strictly decreasing function and is a strictly increasing function2,

C(X),(Y) = P{X 1(x),(Y) y}= P{(y) y} P{X 1(x),(Y) y}= C(Y)(M(Y)) CX,(Y)(F(1(x)), M(y))= C(Y)(M(Y)) CX,(Y)((1 L(x), M(y))= v CX,(Y)(1 u, v)= v CX,Y(1 u, v)

2and by remembering that P(AC B) = P(B) P(A B).

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When and are both strictly decreasing functions3

C(X),(Y) = P{X

1(x), Y

1(y)}

= 1 F(1(x))G(1(y)) CX,Y(F(1(x)), G(1(y)))= 1 (1 u) (1 v) + CX,Y(1 u, 1 v)= u + v 1 + CX,Y(1 u, 1 v)

3 Dependence Concept

We concluded the last section by the copula invariance property to strictly increasing func-tions. It is worthwhile to be noted that this property is not shared by the well know (multi-

variate) elliptical distribution such as the Gaussian and Student one. Furthermore as notedby Embrechts, Lindskog and Mc Neil (2003) because (...) most random variables are notjointly elliptically distributed and and using linear correlation as a measure of dependencein such situation might prove very misleading.4 Let us recall that for Normally distrib-uted random variable the independence property between random variables is equivalent toa Pearson correlation coefficient equal to zero while this equivalence does not hold if therandom variables fail to verify the normality assumption.

This citation and remark provides us with the scope of this second part of this section,namely, documenting dependence measures between random variables. For an overview ofthat topic the reader should refer to Kruksal (1958) or for the specific application to copulato Schweizer and Wolf (1981). Following the properties of copulas, the more interesting mea-sures will be the ones which can be solely defined in term of copula. Let us first reintroducethe notion of linear correlation since it will be used in the next section concerning the pricingof the credit derivatives.

Definition 3.1 LetX andY follow, respectively, the distributionF andG and jointly followthe distribution function H, then the linear correlation coefficient , for X and Y is definedas

(X, Y) =1

V ar(X)

V ar(Y)

[H(x, y) F(x)G(y)] dxdy

or if we use the fact that u = F(x) and v = G(y),

(X, Y) =1

V ar(X)

V ar(Y)

10

10

[C(u, v) uv] dF1(u) dG1(v)

3and by remembering that P(AC BC) = 1 P(A B) = 1 P(A) P(B) + P(A B).4The interested reader should refer to their example on the correlation between two log-normally distrib-

uted random variables. The authors show there that even when perfectly correlated the linear coefficient forthese random variables is near zero (when one variable has standard deviation 1 and the other 4).

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We clearly see from the last equation that the (linear) correlation coefficient is function ofthe inverse of the marginal distribution. Since usually these marginal distributions are notinvariant under monotonous transformation other measure of dependance are more appro-priate when studying the dependance relationship in copulas. This is also a reason why thePearson correlation coefficient is only able to catch linear relationship between variables.

Let us introduce the definition of concordance since this notion will be used in the definitionof other measure of association (which are scale invariant) between random variables.

Definition 3.2 Let(x, y)T and(x, y)T two observations from a vector(X, Y)T of continuousrandom variables. (x, y)T and(x, y)T are concordant if (x x)(y y) > 0 (x, y)T and(x, y)Tare discordant if (x x)(y y) < 0We can now define a the general notion of a concordance function Q.

Theorem 3.1 Let(X, Y)T and (X, Y)T be independent vectors of continuous random vari-ables with joint distribution function H and H, respectively, with common margins F andG. Let C, C denote the copulas of (X, Y)T and (X, X)T, respectively, so that H(x, y) =C(F(x), G(y)) and H(x, y) = C(F(x), G(y)) and let Q denote the difference between theprobability of concordance and discordance of (X, Y)T and (X, X)T respectively

Q(H, H) = P{(XX)( Y Y) > 0} P{(XX)( Y Y) < 0}then

Q(H, H) = 4 H(x, y)dH(x, y) 1or if F(x) = u and G(y) = v,

Q(C, C) = 4 10

10

C(u, v)dC(u, v) 1Proof: If we denote Pc the probability of concordance then 1 Pc is the probability of

discordance so that Q = 2Pc 1. So that

Q(C,

C) = 2P{(

XX)(

Y Y) 0} 1

= 2E[P{(X x)( Y y) 0|X = x, Y = y}] 1= 2E[2H(x, y) F(x)G(y) + 1] 1 = 4E[H(x, y)] 1

Where we have used in the third line

P{X x, Y y} = P{F(x), G(y)} = H(x, y) F(x) F(y) + 1since X and Y are independent with F(x), G(y) uniform random variables. While the defi-nition in term of copula holds by the Sklar theorem.

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According to Scarsini (1984), a set of desirable properties for a concordance measure wouldinclude those following.

Definition 3.3 A numeric measure of association between two continuous random vari-able X and Y whose copula is C is a measure of concordance if it satisfies the followingproperties:

1. is defined for every pairs X, Y

2. 1 1 andX,Y = 13. X,Y = Y,X

4. if X and Y are independentx,y = 0

5. X,Y = X,Y = X,Y6. if C1 and C2 are copulas such that V(C1) < V(C2) thenC1 < C2

7. If{(Xn, Yn)} is a sequence of continuous random variables with copulas Cn and if{Cn}converge point-wise to C, then limn+ Cn = C

A consequence from the last Definition, the Corollary 2.2 and Theorem 2.5 is stated in thefollowing theorem

Theorem 3.2 Let be a measure of concordance for continuous random variable X and Y

1. if Y is almost surely an increasing function of X thenX,Y = M = 1

2. if Y is almost surely a decreasing function of X then thenX,Y = W = 13. if and are almost surely strictly monotone functions, respectively, on RanX and

RanY, then(X),(Y) = X,Y

The most direct measure of association following the theorem 2.1 is the Kendall which isthe difference between the probability of concordance and discordance as previously defined.

Definition 3.4 LetX and Y follow jointly the bivariate distributionH, the Kendalls forX and Y is then defined as

(X, Y) = 4

H(x, y)dH(x, y) 1

or if we use the that u = F(x) and v = G(y),

(X, Y) = 410

10

C(u, v) dC(u, v) 1

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We now move to another measure of association between random variable, the Spearmans.

Definition 3.5 Let(X, Y)T

, (X, Y)T and(X, Y)T be independent copies, the Spearmansfor a random vector(X, Y)T is then defined as

s(X, Y) = 3(P{(X X)(Y Y) > 0} P{(X X)(Y Y) < 0})s(X, Y) = 12

[H(x, y) F(x)G(y)] dF(x) dG(y)or if we use the that u = F(x) and v = G(y),

s(X, Y) = 1210

10

[C(u, v) uv] du dv

Following the last two definitions about the Spearman s and Kendall we have thistheorem

Theorem 3.3 If X and Y are continuous random variables whose copula is C, then Spear-man s as in definition 2.5 and Kendall as defined in definition 2.4 satisfy the measureof concordance definition and theorem 2.2 for a measure of concordance.

Proof: For the measure of concordance definition: the first condition is satisfied by thedefinition of a probability; the second, by the Frechet-Hoeffding bounds; the third, by theexchangeability of copula (ie C(u, v) = C(v, u)); the fourth, by definition of the productcopula;the fifth by the the exchangeability and the Frechet-Hoeffding bounds; For the sixth

and seventh the reader should refer to Nelsen (1998), pg 137.

For the Theorem 3.2: If we take two random variables X, Y and fix X = Y so that theircopula will be the upper Frechet-Hoeffding bounds so that (X, Y) = 4

10 xd(x) 1. We can

use the same argument for the condition 2 and setting X = Y. The third condition holdsby the copula invariance to strictly increasing function property.

Another interesting measure of association is the one related to the tail dependence. Broadlyspeaking with this measure we try so see how random extreme events from different marginaldistribution happen together. Such measure has an implicit interpretation in finance: theprobability two firms default together, the probability two stocks crash together, etc. . .

Definition 3.6 Let (X, Y)T be a vector of continuous random variables with marginal, re-spectively, F and G. The coefficient of upper and lower dependence (if they do exist) arerespectively defined as

limu1

P{Y > G1(u)|X > F1(u)} = U

limu0

P{Y G1(u)|X F1(u)} = L

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The random variables are then said to have upper tail dependance ifU (0, 1] and lowertail dependance ifL (0, 1].

By using the Bayes Theorem (and the remark on footnote 3), the previous expressions canbe redefined in term of copula

Corollary 3.4 Let u I and C be a 2-copula, then the coefficient of upper and lower canbe, respectively re-expressed as

limu1

C(u, u) 2u + 11 u = U

limu0

C(u, u)

u= L

Since the coefficient of upper and lower dependance belongs to the unit interval I and can beexpressed in term of copula, they agrees with the notion of numeric measure of associationas described by Scarsini (1984).

Quite interestingly as noted by Schonbucher (2003) if random variables have tail dependenceit means that there should be some singularities in the volume defined by the shape of the

joint distribution. Namely for the lower dependance case, as u 0, the joint distributionprobability mass or the volume described by the rectangle [0 , u] [0, u] tend to zero at speedL (and not u

2).

4 Copulas families

While having defined the properties of copulas and the notion of measure of association inthe last two sections, this section will be devoted to an overview of some of copula functionsin term of their form and properties since these copulas function will be used in the nextsection devoted on an application to the credit derivatives pricing.

4.1 Elliptical Copulas

The first two copula functions presented in this sub-section come from the class of ellipticaldistributions. Broadly speaking when an elliptical copula (or a joint elliptical distribution) isseen from above, the contour lines of this distribution have elliptical shapes. These copulashave the radial symmetry property and their main advantage is the ease of sampling fromthem while, on the other hand, they do not have a simple closed form.

Definition 4.1 Let denotes the standard univariate normal distribution function and letR the standard normal multivariate distribution with covariance matrix R. The Gaussian

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is not as straightforward as the Archimedean copulas (which will be covered in the nextsection). The derivation which follows rely on the results from Embrechts, Lindskog andMcNeil (2003).

If we use the definition of the coefficient of tail dependance as defined in the previoussubsection apply to it the LHopital rule and remark that P{V v|U = u}=C(u, v)/u(and P{V > v|U = u}=1 C(u, v)/u). We have that

U = limu1

2 + C(u, v)

u

u=v

+C(u, v)

v

v=u

U = limu0

[P{V v|U = u} + P{U v|V = u}]

Since the copulas are exchangeable (i.e C(u, v) = C(v, u)), we have:

U = 2 limu1 P{V u|U = u}If we now define x = F1(u) and y = F1(v) where x, y R with F and G the marginal

distribution of X and Y. We can now rewrite the the previous limit as

U = 2 limx+[P{F

1(V) x|F1(U) = x}] = 2 limx+P{X > x|Y = x}

If F = , the standard Gaussian distribution, and by using the fact that for bivariatestandard Gaussian distribution Y|X = x N(x, 12). We can rewrite the previousexpression as

U = 2 limx+

1 x x1 2 = 2 limx+ 1 x1 1 +

When < 1 the Gaussian copula has no upper tail dependance. By the radial symmetry,this argument also holds for the lower tail dependance coefficient.

If F = t, the Student distribution with degrees of freedom and by using the fact(see Demarta and McNeil (2004) or Galiani (2001) for a formal proof) that P{X >

x|X = x} = 1 t+1

+x2

+1

1/2 (xrx)1r2

. When computing two times the limits of this

last expression we find the coefficient of upper dependance

U = 2 2t1((+ 1)(1 r))1/2

1 + r

This last expression shows that the tail dependance parameter is function of the degree of

freedom and the linear correlation. Let us remark that even with a correlation coefficientequal to zero there is still some tail dependance. While when the degrees of freedom tends tothe infinity, the behavior of tail dependance in the Student distribution tend to the behaviorof the tail dependance in the Gaussian distribution (ie is equal to zero when < 1).

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4.2 Marshall Olkin Copula

The next class of copulas presented in this section was first introduced by Marshall and Olkin(1967a,b) and is particularly relevant when modeling the joint distribution of objects lifetimewhen these lifetime are related to each others. For example the lifetime of light bulbs (from asame brand) or the lifetime of some bonds from companies in a same business sector5, etc. . . )To be more precise, the Marshall-Olkin of copula aim to build multivariate distributionof marginally distributed exponential random variables. The dependance between theseexponential random variables, for example in the bivariate case, is created by taking intoaccount that at any time t during the object lifetime, either one object will die or the twotogether.

In this framework it is understood that the lifetime of an object follows a stopped Poisson()process (ie this object will die at the time of the first jump in the Poisson process).

In the bivariate case (we have objects 1 and 2, with lifetime X and Y), at each time t(before the the death of a component) 3 types of events events (and their complement) canhappen: either only component 1 die (let denote the time when this event happens E1) oronly component 2 die (E2) or component 1 and 2 both die together (E12)

6.

To model these three events, three independent Poisson processes with parameter 1,2,12are used. The survival probability, F for the object 1 at time x (in the bivariate case) is

F1(x) = P{E1 > x}P{E12 > x}

= exp[(1 + 12)x]= exp[1x]

with 1 = 1 + 12. In the bivariate example, the survival function H is defined as

H(x, y) = P{E1 > x}P{E2 > y}P{E12 > max(x, y)}

= exp[1x 2y 12 max(x, y)]

In order now to express the joint survival distribution H, in term of its survival copula, C,we need to work on the last equation. By first noticing that max(x, y) = x + y min(x, y),we can rewrite the last equation,

H(x, y) = exp((1 + 12)x (2 + 12)y 12 min(x, y))= F1(x)F2(y) min(exp(12x), exp(12y))

and by setting

F1(x) = u , F2(y) = v , 1 = 12/(1 + 12) , 2 = 12/(2 + 12)

5In this case the lifetime is understood the time before the bond default.6Let us remark that with this definition the lifetime of component 1, X is min(E1, E12) and the lifetime

of component 2 is min(E2, E12).

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So thatexp(12x) = u

1 , exp(12y) = v2

Now substituting in the definition of H, the survival copula, C, for X and Y is

C(u, v) = uv min(u1, v2) = min(vu11, uv12)

This last computation provides us the form of the Marshall-Olkin copula

CMO1,2(u, v) = min(vu11, uv12) =

u11v, u1 > v2 ,v12u, u1 < v2 .

With 1, 1 (0, 1). The Frechet-Hoeffding bounds in this case are defined as follows

C1,0 = C0,2 = C , C1,1 = M

While we have the following representation for the Spearmans and Kendall (for a formalproof see Nelsen (1998), pg 131)

s(CMO1,2) =

31221 + 22 12 ; (C

MO1,2) =

12

1 + 2 12The coefficient of upper dependence can easily be computed by once again using its definition

in term of copula and applying to it the Lhopital rule,

MOU = limu1

CMO1,2(u, u)

1 u=

2, if1 > 2 ,

1, if1 < 2 .

The coefficient of upper dependance is

CMO1,2

U = min(1,2)

While it was easy to sample from elliptical copulas, for this last copula this becomes moretedious. It is mainly due to the large number of random variable needed to sample: 2n 1uniform random variables in order to later build n dependent exponential random variables.Once the uniform random variables are sampled, one should find an ordering and group themtogether into subsets7. After these subsets are defined one need to use them to compute the

n diff

erent (correlated) intensities,i , of the n objets lifetime. The marginal distribution cannow readily be computed by using the intensities, i . The n variate from the Marshall-Olkin

copula , (i, . . . . , i), are then found by applying the same transformations on the intensitiesand find the parameters is as described in the bivariate case above. The n-Marshall-Olkincopula will now have n parameters.

7For example ifn = 3 then i = 7, one could set the three first random variables be the intensities of thelifetime of the three object; then set the three following be the intensities of the event at which two objectswill die together and the last random variable the the intensity for the time at which all the three objectswill die together.

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4.3 Archimedean Copulas

The class of elliptical copulas is appealing for their ease of sampling while on the other handwe have seen that the measure of dependance computations are quite involved and this classof copulas has restricted properties. It is then natural to try to find other class of copulaswhich could have some desired properties in term of the measure of dependance and whichcould easily be computed. The goal of this section is to present such kind of copula. For anoverview on this specific topic one should refer to Genest and McKay (1986b).

We have seen previously that the product copula was easily computed since this copula isthe product of uniform random variables. If we take the log transformation of this copula,we have

log C(u, v) = log(u) + log(v)

If we are now interested to build a copula not for independent uniform random variables butuniform variables that might be related (either trough a linear or non linear relationship) wecould apply a simple parametric transformation to the product copula in such a way that theparameters in this transformation would produce the desired dependance structure betweenthese uniform random variables. In that case we would have

(C(u, v)) = (u) + (v)

If we now solve the previous equation for C(u, v) we have constructed the following Archimedeancopula

C(u, v) = [1]((u) + (v))

where is called the generator of this Archimedean Copula and [1] is its pseudo-inverse.

Let us now more formally introduce the notion of pseudo-inverse of the previously definedgenerator function by imposing some condition on it so that the Archimedean copula followsthe definition of copula.

Definition 4.3 Let be a continuous, strictly decreasing function from I to [0, +] suchthat (1) = 0. The pseudo inverse of is a function, [1], with Dom[1] = [0, +] andRan[1]=I given by

[1](u) =

u, 0 u (u);0, (u)

u

+

.

Let us remark that if (0) = + then 1=[1] and in the case is called a strictgenerator.

Theorem 4.2 Let be a continuous strictly decreasing function from [0, 1] to [0, +] suchthat(1) = 0 and let[1] be the pseudo-inverse of . let C be the function from [0, 1]2 to[0, 1] given by

C(u, v) = [1]((u) + (v))

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Then C is a copula if and only if is convex

Proof: Nelsen (1998), pg 91.

There as many families of Archimedean copulas as there are function which verifies theprevious definition. We will focus in the next section on (only) one of them but the resultsprovided there can easily be applied to construct other Archimedean copulas.

To sample from Archimedean copulas we will use the algorithm provided by Marshall andOlkin (1988). Since they use the the Laplace transform as a generator function of theArchimedean copula, Let us first define the Laplace transform of a random variable

Definition 4.4 LetY be a random variable satisfying the condition Y > 0 then its Laplacetransform is

Y(s) = E[exp(sY)]with Y > 0

Y(s) is a generator since it is a strictly decreasing, convex (by the Jensen inequality) andinvertible function.

Theorem 4.3 LetF and G be univariate distribution functions with G(0) = 1, then

Hi(xi) =0

F(xi)dG()

is a distribution function

Proof: This holds since, with the Laplace transform of G,

Hi(xi) = ( log F(xi)) = E[exp(( log F(x)))] =0

F(x)dG()

We can see that RanHi=I and Hi is a nondecreasing function.

We can now extend the previous theorem in the multivariate context, with H now a mul-tivariate distribution and Fi the marginal distribution of Xi. If we make the conditionalindependence hypothesis: given , XiXj i = j 8. Then

H(x1, . . . , xn) =

. . . n

i=1

Fi dG()

= (1H1(x1) + . . . + 1Hn(xn))

We have now found back the definition of an Archimedean copula in term of its generatorfunction, .

8This assumption of conditional independence of marginal distribution on a factor is particularly attractivebecause of the product copula ease of computation. This method will be more developed in the next section.

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Definition 4.5 Marshall-Olkin (1988) sampling algorithm for Archimedean copulas

1. Generate a vector (1, . . . , n) form G

2. Generate a vector (u1, . . . , un) from U, a uniform distribution

3. Define xi = F1(u1/yii ), with F(xi) = exp(1Hi(xi)) for i [1, n] and the Laplace

transform of G

Then (x1, . . . , xn) follows the distribution H

4.3.1 Clayton Copula

This class of Archimedean copulas was first introduced by Clayton (1978) from his studieson the epidemiological chronic diseases. His paper is devoted on the estimation of patientdiseases intensities when these intensities are related for patient from as same family (moreprecisely when the patients considered are fathers and their sons). Clayton (1978) onlydeveloped a general form (without imposing any parametric constraint) while Oakes (1982)refined it in term of its parametrization. The Clayton copula has the following form

CClayton (u, v) = max([u + v]1/, 0)

and his generator is defined as follow

Clayton (t) = (t 1)With the parameter [1,) governing the dependance as follow

CClayton1 = W; CClayton0 = ; C

Clayton = M

While the Kendalls is +2

(for a formal proof see Nelsen, pg 131) the Spearmans rho hasto be computed numerically.

It can be shown that the Clayton copula has only lower tail dependance(iif > 0) ascomputed by using the definition of lower dependance and the Lhopital theorem

ClaytonL = limu1

1 2u(u

+ u

1)

1 u = 0

ClaytonL = lim

u0(u + u 1)

u= 21/

To sample from the Clayton copula we will use the Marshall-Olkin algorithm and set G = Ya Gamma(1/) distributed random variable with Laplace transform and Laplace transform

inverse, respectively 1/(s) = (1 + s)1/ and [1]1/ (s) = s

1. This last expression is the

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Clayton copula generator that we have found before. From the Marshall-Olkin algorithmlast step (and the previous definition for Y)

Hi(xi) = 1/( ln(F(xi))) = 1/( ln(u1/yii ))= (1 ln(u1/yii ))1/

If we denote vi = H(xi) then vi CClayton .

5 Application: Copula in Credit Derivatives Pricing

In this section we will be concerned about the default modeling and how to apply these

models to the pricing of credit derivatives products. More precisely the copula methodologydefined in the previous section will help us now to model the dependency between default9

and help us on the pricing of default-correlation derivatives product. This is especiallyrelevant since it is a well known fact that defaults appear in clusters and so cannot betreated as independent random events (de Servigny and Renault (2002)).

5.1 Default modeling

Before modeling the dependance between defaults let us first introduce the framework ofthis section by focusing on the modeling of a single entity default time. We will particularly

rely on the results of Schonbucher and Schubert (2001), Rogge and Schonbucher (2003), sincetheir results are the most general possible by still capturing the necessary characteristic ofwhat a sound default risk model should posses.

5.1.1 Intensities and survival probability

We have seen in the last section that the Marshall-Olkin and Clayton copulas relied onindividual intensities, we will focus here on the company default intensities.

Let us more formally introduce the intensity in the default modeling framework.

Definition 5.1 Let(,Ft[0,T], Q) be a filtered probability space, withF the filtration undera risk neutral probability, Q10. Let N(t) = 1t with a stopping time and M(t) denote the

9Or if we want to be more precise that, if default between two entities are correlated, then the default ofone company will increase the default probability of the one still alive. The most extreme case arises whenlooking at bonds from different maturities but from a same company and in that case the default correlationis 1, so that when one bond default the other default immediately.

10We will assume here that at least one risk neutral measure exists, while usually in the credit marketmany arise since these are function of the recovery assumptions.

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compensated processM(t) = N(t)A(t) t [0, T]

N(t) has a nonnegative intensity process, g(t) under Q if the compensator A(t) can berepresented by

A(t) =t0

g(s)ds t [0, T]

N(t) can then be understood as an inhomogeneous Poisson process which is stopped at itsfirst jump.

Theorem 5.1 Using the previous definition, M(t) follows a (local) martingale under Q

Proof: Let0

s < t. By definition of a Poisson process, N(t) has independent increments

so that N(t)N(s) is independent ofF(s) and has expected value ts g(u)du we have,E[M(t)|F(s)] = E[M(t)M(s)|F(s)] + E[M(s)|F(s)]

= E

N(t)N(s)

ts

g(u)du

F(s)

+ M(s)

= M(s)

We will now link the the default intensity (which can be interpreted as the instantaneousdefault probability) definition to the default probability.

Theorem 5.2 LetQ be the risk neutral probability given byFt that the jump in the processN has not yet occurred be defined as

P(t, T) := Q[ > T|Ft]

Then given some regularity condition (in term of differentiability and measurability11), theintensity of the process N is given by

dA(t)

dt= g(t) = P(t, T)

T

T=t

The main consequence from this last Theorem is that the intensity12 can be recovered fromthe risk neutral survival probability.

Before more formally introducing the assumptions about the default triggering we need tomake some precisions about the filtration, F, in which the model live. Ft[0,T] will be divided

11These conditions are beyond the scope of this paper, the interested reader should refer to Sch onbucherand Schubert (2001), pg 11-12.

12The default intensity at time t is equivalent to the notion of hazard rate when this one is evaluated attime t.

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into two partitions. Dt[0,T] will contain the information only about the occurrence of defaultof the entities in the economy while Gt[0,T] will contain all the other information (which arenot related to the default occurrence). So that if we have n obligors in the economy we haveDt := i[1:n]Dit and Ft := Dt Gt and Fit := Dit GtWe will now focus on the default mechanism. The default of an entity will occur at the time

of the first jump of its Poisson process or equivalently at the first time a default countdownprocess falls between a given barrier.

Definition 5.2 The time of default of an entity i is defined as

i := inf{t : exp(t0i(s)ds) Ui}

with

exp( t0 i(s)ds) is called the default countdown process The default trigger variables Ui are random uniform variables

13 .

i(t) is called a pseudo-intensity14and is a strictly positive stochastic process which is

adapted to Gt

This definition is equivalent to the previous one15. So that the process Ni(t) with t [0, T]as previously defined for the obligor i will generate Dit.

For the Monte Carlo aficionados, this default time definition is convenient since it provides

a direct scheme to simulate default times16

. If we now go back to the default time definitionin term of a stopped Poisson process, we can define the obligor i survival probability at timet until T as,

Pi(t, T) := EQNi(t)|F

it

with Ui as previously defined and by using the Bayes theorem now have that

Pi(t, T) := EQ

exp( T0 i(s)ds)exp( t0 i(s)ds)

Fit

= EQ

exp(Tt

i(s)ds)

Fit

On a pragmatic note; we will assume, from now on, that we can find a measure Q underwhich we calibrate these individual default intensities and survival probabilities from the

market prices of bonds or credit default swap17

.13The information generated by the knowledge ofUi is independent at time t ofGt: (Ui)Gt.14This denomination comes from the fact that i(t) coincides with the intensity gi(t) of obligor i only on

the filtration Fit or if defaults occur independently in the economy. In general it is not an intensity; for anintuition refer to footnote 9.

15since for uniform random variable, Ui, P(pi Ui) = pi16One default path is simulated as follow: one should use the risk neutral survival probability (at different

time) and simulate for each of these times a random uniform variable. The company will default the firsttime the (observed) survival probability is smaller than the simulated random variable.

17Either non parametrically, by bootstrapping the spread curves or parametrically, by applying a non-

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5.1.2 Default dependencies

While the last part of this section was concerned with individual intensities and individualsurvival probabilities we will now extend our analysis to the take into account the defaultrelationship among entities in the economy. By using the previous sections results in termof copula families and time to default and by using the factor modeling as introduced byVasicek (1987) and the conditional independence we have

P(0, 1 > t, . . . , n > t|F(t) (Vt)) =n

i=1

EQNi(t)|F

it (Vt)

with V is a random variable following a given distribution funtion18 and Vt is F(t) measurablefor t [0, T]. The unconditional survival probability is then found by intergrading out thedensity of V. The main advantage of this approach is the ease of computation: conditional

on a factor, the copula to compute is the product copula19

.

(i) Gaussian copulasA Convenient way to take into account the default dependencies in the Gaussian copula(and in the Student copula also) is to use the factor model as developed by Li (2000). Thismodeling can be understood as the conciliation between the copula approach and the firmvalue approach of Merton (1974), as in this framework a firm will default when its assetvalue-like stochastic process, X, fall below a barrier. So for the obligor i we have thefollowing representation of its stochastic process

Xi = iV +

1 2i i

Where V andi are standard independent gaussian random variables (and thus, so is Xi)and Cov(i,j)=0 i = j.

Xi can be interpreted as the value of the asset of the company i and V is interpreted as thegeneral state of the economy. The default dependence between companies in the economycomes from the the factor V and the parameter i (Cor(Xi, Xj) = ij). So, unconditionally(on V) the stochastic processes are correlated but conditionally they are independent.

Since we now have defined the stochastic process, Xi and we can observe the default prob-ability, Fi, of obligor i from the market prices we can now defined its default barrier, Bias follow: Fi(t) = Q{Xi(t) < Bi(t)} = (Bi(t)) so that Bi =

1(Fi(t)). We now define theconditional probability of default of obligor i as

pi|Vt = 1(Fi(t)) iV1 2i

linear optimization on the spread curve as in Nelson and Siegel (1987). For an overview of the techniquesavailable see Schonbucher (2003).

18More precisions concerning the the distribution function of the parameter V will be made in the nextsubsection.

19 One could adopt another method which relies on the default time simulation. In this case the default

time is defined as: i = F1

(ui) where ui is a random variable sampled from your favorite copula and F(0, t)is implied from the observed market prices.

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(ii) Student copulasIn a similar way, we can transform the Gaussian stochastic process, Xi, into a Student t(with degrees of freedom) process, Yi, by using the fact that if X is a standard gaussianrandom variable and Z is a Chi square (with degrees of freedom) random variable then

Xi =/Z

iV +

1 2i i

So the probability of default is now defined as

pi|V,Zt =

t1 (Fi(t)) iV/Z

1 2i

(iii) Double t copulasThis type of modeling has first been introduced by Hull and White (2004); it can be seen asa mixture of the two previous copulas, since now both the idiosyncratic and common latentfactor are Student random variables

Xi =V 2V

1/2V +

1 2

2

1/2i

V and i are both independent Student t random variable with respectively V and degreesof freedom, the distribution of Xi, H, is not known in closed form and so has to be evaluatednumerically 20. The conditional survival distribution is then

pi|Vt = tV

H1(Fi(t)) V2V 1/2 V1 2

2

1/2

(iv) Marshall-Olkin copulasAs previously mentioned in this copula the dependence between default comes from thepossibility of a joint default between companies in the economy. In setting we will definedthe individual obligor factor as as follows

Xi = min(V, i)

Where V, i are independent exponentially distributed random variables with parameter, 1 (with (0, 1)) and so, Xi is an Exponential(1) random variable. Xi can beinterpreted as the obligor i default time. As previously, V, is the factor representing thestate of the economy and i is an individual obligor factor. The dependance between de-fault comes from the factor V which is common to all Xi. We have here circumvented thetractability problem of the Marshall-Olkin by only introducing one factor of dependance

20This will be done in the next section by using the Gaussian kernel.

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and thus in this case we will have only n + 1 independent default intensities21 producingn dependant intensities. The conditional probability of survival of an obligor i at time t isthen

1pi|Vt = Q{i > t}Q{V > t}= 1{V >t}E

1{i>t}

= 1{V > lnFi(t)}Fi(t)

1

Where we have used the independence between i and V and the fact that Xi follows anExponential(1) so that Fi(t) = exp(t), while i follows an Exponential(1 ) and theinvariance of copula under strictly increasing function.

(v) Clayton copulasIn this setting the obligor i factor is defined as

Xi = 1/

ln(i)

V

= ln(i)

V+ 11/

with , the Laplace transform of a Gamma(1/) distribution; i a uniform random variableand V a Gamma(1/) distributed random variable. We can see Xi as the default (or survival)probability which is function of an idiosyncratic and general factor.

If we now make use of of the definition of the default mechanism, and the Marshall-Olkin(1988) sampling algorithm.

1pi|Vt = Q{Xi Fi(t)|V}= Q{(ln

1/Vi ) Fi(t)}

= Q{i exp(V1(Fi(t)))}= exp(V1(Fi(t))) = exp(V(1 Fi(t)))

Where, in the last line, we have used 1=11/ the Clayton copula generator22. Of course

the same technique applies to the other and so other Archimedean copulas.

On a final note let us remark that the one factor model developed in this section caneasily be extended to a multi-factor setting if one wants to model more precisely the defaultcorrelation between obligor.

5.2 CDO pricing

A CDO can be seen as a portfolio of defaultable fixed income products which is decomposedinto several tranches which bear different risk according to their place in the CDO structure

21Instead of 2n 1 intensities in the case where all the possible dependance between obligor default aremodeled. The model with one factor used in this section is quite extreme since it does allow a possibledefault apocalypse where all companies default at a same time V.

22Equivalently the probability of default can be defined as exp(V(1 Fi(t)))

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(the higher is the tranche, the lower is its risk). Each tranche is defined by its attachment anddetachment point, the percentile in the CDO capital structure at which the tranche startssuffering loss and percentile at which the tranche has defaulted. For an introduction aboutthis topic one should refer to OKane et al (2003), Schonbucher (2003) or Gibson(2004).

For example, let consider the first tranche to bear loss in the CDO structure which is calledthe equity tranche. This tranche has 0% attachment point and detachment point 3%. Letassume there are 100 obligors in the CDO pool and each of them as a unit notional (so thatthe CDO pool has a 100 notional). As soon as one obligor defaults in the CDO pool twothings happen for that tranche. Firstly, the buyer of this tranche must provide a defaultpayment to the CDO seller (which is equal to the defaulted obligor notional if we assumezero recovery, this amount is 1 in our example). Secondly, the tranche notional is reducedby the defaulted obligor notional. In this example the investors in the equity tranche willreceive, after the default, a spread on a notional of 2 (and not 3 has previously). In our

example, when more than three obligors have defaulted, the equity tranche has also defaulted(the investors in that tranche have lost a notional of 3 and wont receive any more interestpayment). Now the defaults will affect the second tranche which has the same attachmentpoint than the equity tranche detachment point. This mechanism will continue until theCDO maturity

The main quantity needed to price a CDO is the expected loss in the CDO pool (under therisk neutral measure Q) which is function of the individual obligors default risk within thispool and the dependance between their default time. The copula method presented in thispaper provides a direct way to compute this quantity.

The first step to compute this expected loss in the CDO pool is to compute the conditionaldefault probability for all the individual obligor as described in the previous section. Thengiven a state of the factor these default probabilities are independent so that we can computethe conditional default distribution in the CDO pool at a time t, DV=vt (x). The unconditionaldefault distribution is then

Dt =(V)

DV=vt f(v)dv

with f the density of the factor V. There are several approaches to compute the conditionaldistribution of default in the CDO pool at t, DV=vt (x). A first one is the Fourier transform(Laurent and Gregory (2003)) another is the recursive algorithm (Andersen, Sidenius andBasu (2003), Gibson (2004)) and a last one is the Binomial and Poisson approximation (De

Prisco, Iscoe and Kreinin (2005)). We will use the latter one because of its small computationtime. In this method the number of default in the CDO at a given time and state of thefactor variable is Binomially distributed if all the obligors have the same intensities. If thedefault intensities are not the same among the obligors the Poisson(t) approximation of abinomial distribution can be used by setting t =

ni=1p

it.

If all the obligors have the same notional amount in the collateral, the expected loss on theCDO pool, EL can straightforwardly be computes. Bearing in mind that a CDO tranchewill only take a portion of the CDO pool loss as previously described and by denoting Aj

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and Dj, the tranche j attachment point and detachment point, we have the expected losson that tranche defined as at time t

ELjt = E

Q

[Lt Aj]+

EQ

[Lt Dj]+

In order to compute the expected loss if the obligors have different notional we need toidentify which obligors have defaulted. Following the approach of De Prisco, Iscoe andKreinin (2005), we need first to group the obligors which have the same notional, Nj intosome subsets, kj. Then we can define the probability of a loss, Nj(with mini(Ni) Nj maxi(Ni)),

Q[N = Nj| #default = 1] = l(N = Nj) =n

i=1

1{ikj}ii

,

This last formulation can be extended when more than one issuer has defaulted by not-

ing that probability of having a loss, L given m defaults is approximately the probabilitythat the sum of m independent random variables which have distribution l is equal to L.The distribution of this sum of m random variable is then found by computing the m-foldconvolution of l with itself.

It is worth to be noted that within our setting we, in fact, model the CDO loss process asan inhomogeneous compound Poisson process. Furthermore by adding the in-homogenouscollateral assumption the distribution of the Poisson process increments is allowed to be timevarying.

As we have seen in a CDO tranche investment there are two flows of money. The firstone, the spread payment, which is a discrete super-martingale goes from the CDO issuer to

the CDO tranche investor until the tranche has defaulted. The second one, the defaultpayment, which is function of the loss process is a (discounted) pure jump process goes fromthe CDO tranche investor to the CDO issuer in case of an obligor default in the CDO pool(also until the tranche has defaulted). By using the no arbitrage argument, the differencebetween these two flows should be equal to zero. More formally (assuming a constant risk-freerate) we have

sjTi=1

B(0, ti)[(Dj Aj) ELjti]T0

B(0, s)dELj(s) = 0

with sj the annualized spread on tranche j, B(0, t) the discount factor, i an accrual factor

(which is approximated by the payment frequency per year made from the CDO issuer tothe CDO investors). This last expression is referred as the Mark To Market (MT M) for theCDO investor. By solving for sj, we have

sj =

T0 B(0, s)dEL

j(s)Ti=1 B(0, ti)[(Dj Aj)ELjti]

Let us conclude this section by a note on the CDO tranche hedging with the factor copula.The setting presented in the last section provides a direct way to hedge the CDO tranches

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to the individual credit obligors credit risk. The sensitivity of a tranche j price to a singleobligor i change in default risk (ie a change of one basis point of the CDS spread of obligori at time t is defined as

ji =

MT Mj M T Mj1bps

Tt B(t, s)Pi(t, s)ds

where M T M stands for the CDO issuer Mark To Market (ie the opposite of the CDO investorM T M) and M T M the MTM after the spread widening provided by the copula (and levelof correlation) chosen and Pi is the obligor i survival probability. This procedure need to berepeated for all the obligors in the CDO pool. The behavior of these deltas is particularlyinfluenced by the level of correlation used23.

5.2.1 Numerical example

In this section we will use the tool previously developed by comparing the price given bycopula presented earlier. This section will be an extension to the Kreinin and Iscoe (2005)results by applying them to the Burtschell, Gregory and Laurent (2005) ones.

We will price the tranches (with respectively 0, 3 and 10 percent attachment points) of a5 year maturity synthetic CDO with a collateral composed of 40 different obligors havingCDS spreads ranging (uniformly) from 60 to 150 basis points with a 40% recovery rate.Furthermore we assume their term structure is flat so that we use the credit triangle torecover their hazard rate24. The notional of these obligors range between 1 and 5 and theCDO notional is equal to 10025. Finally we make the assumption this CDO pays a spreadquarterly and the risk free rate term structure is flat at 5 percent.

We first use the Gaussian and the Student copula to price the equity tranche and use a fixedcorrelation parameter (ie i = i).

2 0 0.2 0.4 0.6 0.8 1Gaussian 3958 2492 1677 1106 747 123Student(12) 3113 2158 1523 1050 674 205Student(6) 2596 1890 1381 980 636 241

Table 1: Equity tranche (0%-3%) prices as function of the correlation and copula

We can see from this last table that even tough the price given by the different copulas arenot the same, they follow a same pattern. Namely the equity spread is a decreasing function

23The interested reader should refer to Schloegl and Greenberg (2003) for further discussion about thebehavior of CDO tranche deltas

24Namely the obligor i intensity is equal to si1R

. With R the recovery rate and si the obligor i CDS spread.For more details about the last computation the interested reader should refer to Okane et al (2003), pg 32

25More specifically in the CDO pool there are 5 obligors with a notional of 1, 20 with a notional of 2, 10with a notional of 3 and 5 with a notional of 5. Where the higher is the notional the smaller is the obligorrisk.

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of the default correlation. This is because an increase in the default correlation increasesthe variance of number of default in the CDO collateral distribution. At the extreme, whenthe correlation is equal to 1 there is a high probability that there wont be any default (anda relatively high probability that all obligors will default) so that the equity tranche is lessrisky in a high correlation environment.26

In order to compare the prices given by the different copulas we will now match the pricethey give for the equity tranche to the price given by the Gaussian copula and compare theprice they give for the other tranches.

2 0.2 0.4 0.6 0.8Gaussian 599 561 517 427Student(12) 611 558 482 410Student(6) 628 563 479 400Clayton 614 570 499 408t(4)-t(4) 449 388 317 272MO 312 227 203 205

Table 2: Mezzanine tranche (3%-10%) prices as function of the correlation and copula

We have focused here on the correlation between 0.2 and 0.8 since in our example the Studentcopula cannot reproduce the Gaussian equity price in extreme correlation environment. Wecan see from this last table that the Marshall Olkin copula tend to produce totally differentprices than the Clayton Student and Gaussian copulas; while the double t copula produceprice for that tranche in between. More interestingly even if the Clayton, Student andGaussian copulas tend to give the same price we can see (from the plots in appendix) theydont produce a same shape for distribution of number of default in the CDO pool.

2 0.2 0.4 0.6 0.8Gaussian 11 22 33 42Student(12) 11 23 35 45Student(6) 10 22 35 46Clayton 11 24 37 47t(4)-t(4) 11 23 34 48MO 23 38 49 57

Table 3: Senior tranche (10%-100%) prices as function of the correlation and copula

We can note that the Clayton, Gaussian and Student and double t copulas tend to producethe same prices and the senior tranche price is a negative function of the correlation level27.

26This can be seen from the default distribution in the CDO pool default distribution plot given by thedifferent copulas in appendix.

27Once again we refer to the distribution plots in appendix where we clary see, at high correlation level,a (relatively) high probability mass at the probability distribution extreme right tail.

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6 Conclusion

This paper has presented the building blocks of the CDO pricing with factor copula. Thisapproach to price the CDO is not unique but is considered as the benchmark model becauseof its ease of implementation and the flexibility it does provide. However this method (theone factor coupled with the copula approach) fails to explain the price observed in themarket, as it tends to underprice the more senior tranche, so there is room for improvement.

Several paths are possible if one believes in the copula approach. The first one, would be toimprove this type of modeling by including transition probabilities between the factor statesor by making the copula parameter stochastic (Andersen and Sidenius (2004); Berd, Engleand Voronov (2005) or as Patton (2001)). Another path to further investigate is to use theapproach of Schonbucher (2003) in which he builds a fully dynamic model for the individualobligor default risk but in that case one should come up with a good way to reduce theinformation in order to keep the model tractable to price CDO.

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References

[1] Andersen, L., Sidenius, J. and Basu, S. (2003), All Your Hedges in One Bas-ket, RISK, November, 67-72.

[2] Andersen, L. and Sidenius, J. (2005), Extensions to the Gaussian Copula:Random Recovery and Random Factor Loadings, Journal of Credit Risk, 1.

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A Appendix

x

y

z

Gaussian

Number of default in the CDO pool (y [0, 40]) distribution (z [0, 1]) as function of thecorrelation coefficient 2 (x

[0, 1]) for the Gaussian copula evaluated at year five (with the

same assumptions as in the numerical example).

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x

y

z

Student

Number of default in the CDO pool (y [0, 40]) distribution (z [0, 1]) as function of thecorrelation coefficient 2 (x [0, 1]) for the Student (6) copula evaluated at year five (withthe same assumptions as in the numerical example).

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x

y

z

Double t

Number of default in the CDO pool (y [0, 40]) distribution (z [0, 1]) as function of thecorrelation coefficient 2 (x [0, 1]) for the double t (t(4)-t(4))copula evaluated at year five(with the same assumptions as in the numerical example).

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x

y

z

Clayton

Number of default in the CDO pool(y [0, 40])probability distribution (z [0, 1]) asfunction of the copula parameter (x [0, 1.5]) for the Clayton copula evaluated at yearfive (with the same assumptions as in the numerical example).

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x

y

z

MarshallOlkin

Number of default in the CDO pool (y [0, 40]) distribution (z [0, 1]) as function of thecopula parameter (x (0, 1)) for the Marshall-Olkin copula evaluated at year five (withthe same assumptions as in the numerical example).

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Verschuere Applications to Synthetic CDO Pricing86

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